538 J. FOR. SCI., 53, 2007 (12): 538–547 JOURNAL OF FOREST SCIENCE, 53, 2007 (12): 538–547 e determination of the volume of trees and their parts by means of the basic characteristics such as dbh and height, recommended from the practical point of view, is burdened with errors resulting from variation of the stem form of trees. is variation is a result of differences in the rate of diameter incre- ment at different heights of the stem and differences in the height increment of trees (M 1970). ese differences may be caused by many factors including species variation, climatic factors, site quality, age of trees and stands, defoliation, and stand density (M 1994). e taper of the upper stem section is also affected by the length of the crown (K, V 1981; L 1963; S 2002). Within the crown, stem diameters at particular heights are generally smaller in compari- son with trees of the same dimensions but shorter crowns. Also genetic factors may decide on the stem form. During the study aimed at the provenance va- riation of Abies grandis (S, K 2005) it was found that the stem form variation was influenced by the provenance (genotype). Provenances the par- ent stands of which grew at higher elevations were characterized by greater stem volume than prove- nances from lower elevations, at the same values of dbh and height. In Fagus sylvatica D (2003) found differences between mountain beech and lowland beech in respect of the stem form. Similar conclusions were drawn from studies on the stem form of Picea abies (C 2002; S, K 2004). e knowledge of factors affecting the stem form of forest trees is the basis of correct determination of tree volume, not burdened with systematic errors. Stem tapering, affecting the quality of timber to a certain extent, may be one of the criteria of prov- enance selection. The purpose of this study was to estimate the provenance variation of the tree form factor and ta- per of European larch on the basis of empirical data acquired by measurements of dendrometric char- acteristics of 20 larch provenances tested under the 1967 Polish Provenance Experiment on Larch. e study was carried out in the comparative experimen- tal area established in Krynica (the Beskid Sądecki mountain range, southern Poland) and supervised by the Department of Forest Tree Breeding, Faculty of Forestry, Agricultural University of Cracow. Variation of the tree form factor and taper in European larch of Polish provenances tested under conditions of the Beskid Sądecki mountain range (southern Poland) J. S 1 , M. K 2 1 Department of Forest Mensuration, Faculty of Forestry, Agricultural University of Cracow, Poland 2 Department of Forest Tree Breeding, Faculty of Forestry, Agricultural University of Cracow, Poland ABSTRACT: e genetic variation in 20 provenances of European larch, growing under site conditions of the Beskid Sądecki mountain range (experimental area in Krynica), was investigated during a long-term study carried out within the 1967 Polish Provenance Experiment on Larch. Data consisted of diameter measurements taken outside bark on standing trees of the analyzed provenances. Results showed that there was no distinct variation in the tested larch populations in respect of stem form. Some differences between compared provenances in respect of stem taper and form factor were the result of differences in tree height and diameter. Keywords: genotype; planting experiment; stem profile J. FOR. SCI., 53, 2007 (12): 538–547 539 MATERIAL This study was aimed at 20 provenances of larch from the entire territory of Poland (Fig. 1) tested in the experimental area in Krynica situated in the Carpathian Forest Region (sub-region of the Gorce and Beskid Sądecki mountain ranges). The experimental area is located in the Wojkowa for- est section of the Forest Experimental Station in Krynica at 785 m above sea level, i.e. in the middle part of the lower mountain zone. Its site type was classified as the mountain forest site. Individual provenances were planted in five replications (plots 20 × 20 m each) and distributed following the rule of the “Latin rectangle”. A detailed description of the study area may be found in the author’s earlier paper (K 2001). The study material consisted of dbh measurements of all trees, and height meas- urements of 5 trees in each plot, as well as diameter measurements of stem sections taken on 3 standing trees selected at random for each provenance in 5 replications (15 trees of each provenance). Meas- ured trees were 39 years old. The section diameter measurements were taken at the base of the stem as well as 0.5 m, 1.3 m, 2.0 m above the ground level, and then every 2 m up to the tree top. The last measurement was taken about 2–3 m from the tree top. In total, section diameter measurements were taken on 300 trees. The Ledha GEO laser dendrometer was used. METHODS Because the parent stands of tested provenances of European larch were growing in various regions of Poland (Fig. 1), apart from the variation of the stem form, also the geographical variation was analyzed. For this purpose the provenances were included in five groups depending on the geographical location of parent stands: I – provenances from northern Poland (1, 2, 4, 6); II – provenances from central Poland (7, 8, 9); III – provenances from the Świętokrzyskie Moun - tains (10, 11, 12, 13, 14, 19); IV – provenances from the Sudetes (20, 21, 22, 23, 24); V – provenances from the Carpathians (16, 18). On the basis of section measurements taken on standing trees, diameters at 100 relative heights (0.00, 0.01, 0.02 … 0.99) were computed for each tree using interpolation according the 3 rd degree Hermite’s functions (K 1999). An example of the curve computed by the interpolation method where diameters measured at different heights were joined is shown in Fig. 2. Volumes of the stem as well as of merchantable timber of each tree were computed using a section method with section length equal to 0.01 of the tree length. Volumes were computed using Smilian’s equation. Volumes computed from the sum of vol- umes of individual sections were accepted as real values in further analyses. Fig. 1. Location of parental larch stands of provenances investigated on a test site at Krynica Experimental Forest Station 540 J. FOR. SCI., 53, 2007 (12): 538–547 e estimation of variation of the tree form factor for all data within individual provenances was done in several stages. Since the values of the tree form factor most often depend on the tree size, a direct comparison of form factors of provenances differing in diameter and height may lead to erroneous conclusions (A 1993). In such a case possible differences in the values of the form factor may be a result of differences in the diameter and height of trees of individual provenances. To eliminate these differences a regression model was worked out for all data. is model described the form factor as an independent variable being explained by dependent variables. e model form factors computed from the regression equation were the mean values for given tree dimensions (dependent variables). To find whether a given provenance is characterized by higher or smaller form factor values, the real (computed on the basis of volume, diameter and height of the tree) and the model form factors were computed for each tree. en the differences between model and real form fac- tors were computed. e values of differences between these form factors provided information indicating whether a given provenance significantly differed in respect of this trait from the total population. Analyses of differences between form factor values were carried out for the true (f 0.05 ) and breast height (f 1.3 ) stem form factors. For this purpose regression models describing the relationship between the form factors and the basic biometric characteristics of trees, such as height and diameter, were worked out in order to compare real values with model values of the form factor by computing the absolute (δf) (equa- tion 1) and per cent (δf % ) (equation 2) differences between model (f pred ) and real values (f obs ). δf = f pred – f obs (1) f pred – f obs δf % = –––––––––– × 100% (2) f obs e determined errors assumed to be the basis of the comparison between the stem form factors of various provenances became the basis of the estima- tion of provenance diversification in respect of the stem form factor. e estimation of the stem taper was done on the basis of the coefficient of tapering proposed by K (1944) (equation 3). d 0.1 – d 0.5 z = –––––––––– (3) 0.4h e coefficient of tapering determined in such a way is, however, dependent on tree dimensions, and differences in its value may result from differences in the rate of tree growth of individual provenances (K 2001). To eliminate their influence the co- efficient of tapering z r was used. It was proposed to compute this coefficient on the basis of relative diameters (equation 4). z r = 2.5 × (d r0.1 – d r0.5 ) (4) A detailed analysis of the effect of provenances on the stem profile and taper of tree stems was carried out by the comparison of diameters from relative heights: 0.05, 0.10, 0.20, … 0.90. e effects of the provenance and provenance region on the values of relative diameters were analyzed using the analysis of variance. Stand density is one of the hypothetic factors that may affect the stem form of trees. is is why also the analyses determining the relationship between the variation of the stem form and stand density were carried out. e stand density index (SDI) proposed by Reineke (W et al. 2002; Z 2005) was used. is index is a relative measure of density elaborated for even-aged stands, and it is determined on the basis of the number of trees per hectare (TPH) and the quadratic-mean dbh (d q ) (equation 5). Fig. 2. Diameters measured on the stem and interpolation curve computed using Hermite’s method (h = 18.7 m, dbh = 20.05 cm) 25 20 15 10 5 0 Interpolation Measured diameter d (cm) 0 2 4 6 8 10 12 14 16 18 20 h (m) J. FOR. SCI., 53, 2007 (12): 538–547 541 dbh q 1.6 SDI = TPH ( –––––– ) (5) 25 is index is based on the relationship between the mean dbh and the number of trees per unit area. In order to check whether the density index SDI significantly modifies the variation of the true form factor the method of multiple regression was used with the tested true form factor as a dependent variable and the stand density (SDI), height (H), and diameter from height 0.05h (D 0.05h ) as independent variables. RESULTS Variation of breast height and true form factors Breast height form factor The breast height form factor of the analyzed provenances of European larch turned out to be independent of the values of the basic dendromet- ric characteristics of trees such as dbh, height or crown length (absolute and relative). us, when comparing the breast height form factors of different provenances there was no need to exclude the effect Fig. 3. Mean values of the breast height form factor of European larch of different provenances 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 Breast height form factor (f 1.3 ) Mean Mean ± Standard deviation Mean ± 1.96*Standard deviation 1 2 4 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 24 Provenance Fig. 4. Values of the breast height form factor (f 1.3 ) of Eu- ropean larch depending on the provenance region 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 Breast height form factor (f 1.3 ) 1 2 3 4 5 Region • Mean  Mean ± Standard deviation [ Mean ± 1.96*Standard deviation 542 J. FOR. SCI., 53, 2007 (12): 538–547 of dendrometric characteristics on their variation. e tree form factors of partial populations of Eu- ropean larch under comparison ranged on average from 0.441 for provenance 2 (Pelplin) to 0.493 for provenance 1 (Myślibórz Północ) (Fig. 3). On the basis of the analysis of variance, with the previous test of homogeneity of variance, it was found that the observed differences in the mean values of the breast height form factor of tested provenances were statistically insignificant (α = 0.05). No significant differences were found in the mean values of form factors determined for the different provenance regions of larch. e mean values of form factors for larches from the respective regions ranged from 0.454 for region 2 (central Poland) to 0.464 for region 5 (the Carpathians) (Fig. 4). True form factor (f 0.05 ) In the case of the true form factor f 0.05 the varia- tion between individual provenances was consid- erably greater (Fig. 5). For two provenances, i.e. provenance 1 (Myślibórz Północ) and provenance 6 (Konstancjewo-Tomkowo), the difference was sig- nificant (α = 0.05). e analysis at the region level also showed certain diversification of the true form factor (Fig. 6). e Fig. 6. Values of the true form factor (f 0.05 ) of European larch depending on the provenance 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 Form factor (f 0.05 ) Mean  Mean ± Standard deviation I Mean ± 1.96*Standard deviation 1 2 3 4 5 Region • Fig. 5. Values of the true form factor (f 0.05 ) of European larch depending on the provenance region 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 Form factor (f 0.05 ) Mean  Mean ± Standard deviation I Mean ± 1.96*Standard deviation 1 2 4 6 7 8 9 10 11 12 13 14 16 18 19 20 21 22 23 24 Provenance J. FOR. SCI., 53, 2007 (12): 538–547 543 analysis of variance, carried out in order to compare the mean form factors of individual regions, indi- cated the existence of significant differences in form factor values between the different regions. On the basis of multiple comparisons by Tukey’s test prov- enances from central Poland and from the Sudetes were found to significantly differ in the mean values of the form factor (regions 2 and 4). Using the multiple regression analysis the values of the true form factor were found to depend on the diameter and height of trees. erefore, the observed differences could result from provenance diversifi- cation in respect of tree diameter and height. For this reason a regression model describing the form factor by means of two independent variables, dbh and height, was used to compare the values of form factors of individual provenances. On the basis of the corrected coefficient of determination it was stated that a linear equation (equation 6) describing the relationship between the true form factor and the diameter d 0.05 and height explained about 14% of the form factor variation. f 0.05 = 0.3180 + 0.007657 × h – 0.001846 × d 0.05 (6) e information on the provenance diversification of the true form factor was obtained by comparison of residual values of the regression model. For this purpose in each of 300 trees making up the study Fig. 8. Mean residual values of the equation of multiple regres- sion used to determine the true form factor depending on the provenance region Residuals Mean  Mean ± Standard deviation I Mean ± 1.96*Standard deviation 1 2 3 4 5 Region 0.10 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 –0.08 –0.10 Fig. 7. Mean residual values of the equation of multiple regres- sion used to determine the true form factor depending on the provenance 0.12 0.10 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 –0.08 –0.10 –0.12 Residuals Mean  Mean ± Standard deviation I Mean ± 1.96*Standard deviation 1 2 4 6 7 8 9 10 11 12 13 14 16 18 19 20 21 22 23 24 Provenance • • 544 J. FOR. SCI., 53, 2007 (12): 538–547 material the model form factor was computed and compared with the real one in accordance with equation 1. e residual values of the equation for the form factor computed for individual prove- nances were similar. e analysis of variance showed that the residuals of the regression model for individ- ual provenances ranging from –0.019 for provenan- ce 8 (Rawa mazowiecka) to +0.033 for provenan- ce 1 (Myślibórz Północ) did not differ significantly (Fig. 7). Similar results were obtained when residual values for individual regions were compared (Fig. 8). In this case the elimination of the effect of tree diameter and height caused that differences in the values of the form factor observed for region 2 (provenances from central Poland) and region 4 (provenances from the Sudetes) turned out to be insignificant. No differences were found in residual values of the re- gression equation describing the form factor on the basis of dbh and height. Differences in the values of the true form factor found by the direct comparison were also caused by diversification of dimensional characteristics of trees in this case. Stem tapering The stem tapering determined according to Krenn’s equation (equation 2) showed consider- able provenance diversification. The mean value of taper varied from 0.67 cm/m for provenance 23 (Szczytna Śląska) to 1.00 cm/m for provenance 2 (Pelplin). The extreme differences between the mean stem taper of individual provenances and the mean taper of populations under investiga- tions ranged from –0.15 (provenance 23 – Szczytna Śląska) to +0.17 (provenance 2 – Pelplin) (Fig. 9). The occurrence of groups significantly differing from one another was found on the basis of the analysis of variance. e multiple regression analysis showed that stem tapering was strongly correlated with dbh and height of trees. e coefficient of multiple correlation for this relationship was 0.70. As it was shown by the value of the corrected coefficient of determination 49% of taper variation was explained by dbh and height of trees. Differences between individual prov- enances in respect of tapering values were therefore Fig. 9. Differences between the mean stem tapering of individual provenances determined accord- ing to K (1944) and the mean stem tapering determined for all empirical data 0.20 0.15 0.10 0.05 0.00 –0.05 –0.10 –0.15 –0.20 ZK provenance – ZK population 23 1 20 4 13 22 12 24 18 16 14 6 11 9 21 7 19 8 10 2 Provenance Fig. 10. Differences between the mean stem tapering of individual provenances deter- mined according to K’ (1944) modified equation and the mean stem tapering deter- mined for all empirical data 23 1 13 9 20 14 4 12 22 6 19 24 16 10 11 7 18 21 8 2 ZKM provenance – ZKM population Provenance 0.20 0.15 0.10 0.05 0.00 –0.05 –0.10 –0.15 –0.20 J. FOR. SCI., 53, 2007 (12): 538–547 545 caused to a great extent by differences in the rate of height and diameter growth. e effect of dbh and height on variation of stem tapering was eliminated by computing the relative tapering (equation 4). In this case the range of taper variation distinctly decreased but the extreme mean values were still observed in provenance 23 (Szczyt- na Śląska) and provenance 2 (Pelplin) (Fig. 10). The comparison of means, using the analysis of variance, showed that when the effect of dbh and height was eliminated the mean values of tapering of individual partial populations did not differ sig- nificantly. e stem profile variation Individual provenance regions differed in respect of the range of variation of relative diameters d 0.05 at individual heights of the stem. However, on the basis of a direct comparison of the trait under analysis it was observed that the different regions were charac- terized by similar mean values of relative diameters at individual stem heights (Fig. 11). e differences in mean relative diameters at individual heights were not larger than 0.02. A little greater diversification of average stem profiles occurred between individual provenances. e one-way analysis of variance did not show any differences in the values of mean relative diameters d w0.05 from the particular heights which would have been caused by the provenance or by the provenance region. is was confirmed by results of the analysis of the tree form factors and tapering. Relationship between stand density and variation of the stem profile Using simple linear regression a slight, although statistically significant (α = 0.05) effect of stand density index (SDI) on variation of the true form factor f 0.05 was found (Fig. 12). e coefficient of correlation, and in consequence the proportion of explained variation, was however relatively small since the value of the coefficient of determination (R 2 ) was only 0.015. After using the model of multiple regression in which apart from the index SDI also the relative diameter at height 0.05h (d w0.05 ) and the tree height were independent variables, in the description of variation of the true form factor (f 0.05 ) it turned out that the index of stand density SDI was an insignifi- cant variable. e proportion of variance being ex- Fig. 11. Stem profiles of European larch from individual prov- enance regions Fig. 12. Relationship between the true form factor and the stand density index (SDI) Table 1. Parameters of a multiple regression model describing the true stem form factor f 0.05 on the basis of the stand density index (SDI), height (H) and relative diameter d w0.05 , and estimation of their significance Variables Parameters of a multiple regression equation and estimation of their significance parameter ß standard error of parameter ß t-statistics value probability level Free term 0.30652 0.01906 16.08178 0.00000 SDI 0.00002 a 0.00002 1.24408 0.21446 H 0.00741 0.00122 6.09091 0.00000 d w0.05 –0.00175 0.00062 –2.84490 0.00475 a Parameter insignificant at α < 0.2145 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Relative diameter (d 0.05 ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Relative height 0.56 0.54 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 f 0.05 300 400 500 600 700 800 900 1,000 1,100 1,200 SDI –– 1 –•–2 - 3 • 4 5 546 J. FOR. SCI., 53, 2007 (12): 538–547 plained by this trait did not differ significantly from zero (Table 1). DISCUSSION e analyses of the stem form of European larch, described by means of the tree form factor, or di- rectly expressed by means of the taper or diameters at individual heights of the stem, did not show the influence of the provenance on its variation. Taking into account the dimension traits of trees, the pro- portion of variance explained by provenances did not differ significantly from zero. Results of this study differ from results of the study on Abies grandis (S- , K 2005) which showed that the stem form of that tree species was a trait determined by the genotype. Similar results were also expected on the basis of studies aimed at the stem form of mountain and lowland Fagus sylvatica (D 2003), as well as studies concerning Picea abies stands which showed differences between mountain and lowland stands in respect of the stem form (C 2002). At the present state of investigations it is difficult to make comprehensive hypotheses on the observed regularities in variation of the stem form of the studied larch partial populations. e authors of the present study are of the opinion that their results permit to formulate the hypothesis about specific properties of larch as a species the tree form factor and tapering of which are determined by growth conditions to a greater extent than by the provenan- ce (genotype). However, growth conditions in this case should be understood as conditions on a macro scale. Specific growth conditions on a micro scale, occurring in individual experimental plots of the provenance experiment and determined on the basis of the SDI index, did not significantly affect the form of tree stems. CONCLUSIONS Differences in the values of the stem form and taper, observed on the basis of a direct comparison, resulted from differences in the growth rate of the analyzed larch provenances causing a significant diversification of diameter and height of trees. e values of the breast height form factor ranged on average from 0.441 (Pelplin) to 0.493 (Myślibórz Północ). However, the differences between prov- enances were not statistically significant. In the case of the true stem form factor (f 0.05 ) sig- nificant differences in absolute values of this trait were found between provenances from Myślibórz Północ and Konstancjewo-Tomkowo. However, these dif- ferences resulted from the relationship between the true form factor and diameter and height of trees. e elimination of the effect of diameter and height made these differences statistically insignificant (α = 0.05). More detailed information on the stem form of larch was obtained on the basis of the analysis of rela- tive diameters at different heights of the stem. In this case, irrespective of assumed diameter in respect of which relative diameters at individual stem heights were computed, and in spite of a certain diversifi- cation of mean stem profiles of individual partial populations, no significant effect of the genotype (provenance) on their variation was found. e vari- ation of the stem form was not significantly affected by the provenance region, either. e observed dif- ferences in mean diameters from individual stem heights were statistically insignificant. The results of the analysis of the relationship between the stem form of larch of the tested prov- enances and the index of stand density, obtained dur- ing this study, were not expected. Although there was a slightly positive correlation between the true form factor and the stand density index, it was however, as proved by detailed analyses, the result of differ- ences caused by dendrometric traits of the analyzed provenances. Their elimination showed that the stand density index had no influence on variation of the true stem form factor. e results obtained during this study indicated specific growth properties of European larch of the tested partial populations which cause that the form factor and taper of the stem do not depend on the provenance. It should be pointed out, however, that this study concerned only one of the so called paral- lel experimental areas of the 1967 Polish Provenance Experiment on Larch, i.e. the Krynica experimental area situated in the Beskid Sądecki mountain range. erefore, results of this study need to be confirmed by similar studies carrield out in other experimental areas of different growth conditions. 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Stand density index in uneven-aged ponderosa pine stands. Ca- nadian Journal of Forest Research, 33: 96–100. ZEIDE B., 2005. How to measure stand density. Trees, 19: 1–14. Received for publication July 3, 2007 Accepted after corrections September 10, 2007 Corresponding author: Dr. J S, Agricultural University of Cracow, Faculty of Forestry, Department of Forest Mensuration, Al. 29 Listopada 46, 31-425 Cracow, Poland tel.: + 48 12 662 5011, fax: + 48 12 411 9715, e-mail: rlsocha@cyf-kr.edu.pl Změny stromové výtvarnice a sbíhavosti kmene u modřínu opadavého polských proveniencí ověřované v podmínkách horského pásma Beskyd Sądecki (jižní Polsko) ABSTRAKT: V dlouhodobé studii, která se uskutečnila v rámci Polského provenienčního pokusu s modřínem 1967, jsme sledovali genetickou proměnlivost u 20 proveniencí modřínu opadavého, který se nachází ve stanovištních podmínkách horského pásma Beskyd Sądecki (na pokusné ploše v Krynici). Údaje pocházely z měření tloušťky kme - ne s kůrou na stojících stromech sledovaných proveniencí. Získané výsledky nenaznačily u sledovaných populací modřínu žádné zřetelné změny ve tvaru kmene. Některé rozdíly mezi srovnávanými proveniencemi ve sbíhavosti kmene a stromové výtvarnici vyplynuly z rozdílů ve stromové výšce a tloušťce. Klíčová slova: genotyp; provenienční pokus; profil kmene . University of Cracow. Variation of the tree form factor and taper in European larch of Polish provenances tested under conditions of the Beskid Sądecki mountain range (southern Poland) J. S 1 ,. individual provenances was done in several stages. Since the values of the tree form factor most often depend on the tree size, a direct comparison of form factors of provenances differing in diameter. experiment and determined on the basis of the SDI index, did not significantly affect the form of tree stems. CONCLUSIONS Differences in the values of the stem form and taper, observed on the basis of